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Nonlinear control of unsteady finite-amplitude perturbations in the Blasius boundary-layer flow

Published online by Cambridge University Press:  26 November 2013

S. Cherubini*
Affiliation:
DynFluid Laboratory, Arts et Metiers ParisTech, 151, Bd. de l’Hopital, 75013 Paris, France
J.-C. Robinet
Affiliation:
DynFluid Laboratory, Arts et Metiers ParisTech, 151, Bd. de l’Hopital, 75013 Paris, France
P. De Palma
Affiliation:
DMMM, CEMeC, Politecnico di Bari, Via Re David 200, 70125 Bari, Italy
*
Email address for correspondence: s.cherubini@gmail.com

Abstract

The present work provides an optimal control strategy, based on the nonlinear Navier–Stokes equations, aimed at hampering the rapid growth of unsteady finite-amplitude perturbations in a Blasius boundary-layer flow. A variational procedure is used to find the blowing and suction control law at the wall providing the maximum damping of the energy of a given perturbation at a given target time, with the final aim of leading the flow back to the laminar state. Two optimally growing finite-amplitude initial perturbations capable of leading very rapidly to transition have been used to initialize the flow. The nonlinear control procedure has been found able to drive such perturbations back to the laminar state, provided that the target time of the minimization and the region in which the blowing and suction is applied have been suitably chosen. On the other hand, an equivalent control procedure based on the linearized Navier–Stokes equations has been found much less effective, being not able to lead the flow to the laminar state when finite-amplitude disturbances are considered. Regions of strong sensitivity to blowing and suction have been also identified for the given initial perturbations: when the control is actuated in such regions, laminarization is also observed for a shorter extent of the actuation region. The nonlinear optimal blowing and suction law consists of alternating wall-normal velocity perturbations, which appear to modify the core flow structures by means of two distinct mechanisms: (i) a wall-normal velocity compensation at small times; (ii) a rotation-counterbalancing effect al larger times. Similar control laws have been observed for different target times, values of the cost parameter, and streamwise extents of the blowing and suction zone, meaning that these two mechanisms are robust features of the optimal control strategy, provided that the nonlinear effects are taken into account.

Type
Papers
Copyright
©2013 Cambridge University Press 

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References

Adrian, R. J. 2007 Hairpin vortex organization in wall turbulence. Phys. Fluids 19, 041301.Google Scholar
Alizard, F., Cherubini, S. & Robinet, J. Ch. 2009 Sensitivity and optimal forcing response in separated boundary layer flows. Phys. Fluids 21, 064108.Google Scholar
Bagheri, S. & Henningson, D. H. 2011 Transition delay using control theory. Phil. Trans. R. Soc. Lond. 369, 13651381.Google ScholarPubMed
Barbagallo, A., Dergham, G., Sipp, D., Schmid, P. J. & Robinet, J. C. 2012 Closed-loop control of unsteadiness over a rounded backward-facing step. J. Fluid Mech. 703, 326362.CrossRefGoogle Scholar
Bewley, T. R. & Liu, S. 1998 Optimal and robust control and estimation of linear paths to transition. J. Fluid Mech. 365, 305349.Google Scholar
Bewley, T. R., Moin, P. & Temam, R. 2001 DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms flows. J. Fluid Mech. 447, 179225.Google Scholar
Bottaro, A. 1990 Note on open boundary conditions for elliptic flows. Numer. Heat Transfer B 18, 243256.Google Scholar
Brandt, L., Schlatter, P. & Henningson, D. S. 2004 Transition in a boundary layers subject to free stream turbulence. J. Fluid Mech. 517, 167198.Google Scholar
Cherubini, S. & De Palma, P. 2012 Nonlinear optimal perturbations in a Couette flow: bursting and transition. J. Fluid Mech. 716, 251279.Google Scholar
Cherubini, S., De Palma, P., Robinet, J.-Ch. & Bottaro, A. 2010a Rapid path to transition via nonlinear localized optimal perturbations. Phys. Rev. E 82, 066302.Google Scholar
Cherubini, S., De Palma, P., Robinet, J.-Ch. & Bottaro, A. 2011a Edge states in a boundary layer. Phys. Fluids 23, 051705.Google Scholar
Cherubini, S., De Palma, P., Robinet, J.-Ch. & Bottaro, A. 2011b The minimal seed of turbulence transition in a boundary layer. J. Fluid Mech. 689, 221253.Google Scholar
Cherubini, S., Robinet, J.-Ch., Bottaro, A. & De Palma, P. 2010b Optimal wave packets in a boundary layer and initial phases of a turbulent spot. J. Fluid Mech. 656, 231259.Google Scholar
Chevalier, M., Hoepffner, J., Akervik, E. & Henningson, D. S. 2007 Linear feedback control and estimation applied to instabilities in spatially developing boundary layers. J. Fluid Mech. 588, 163187.CrossRefGoogle Scholar
Chevalier, M., Hogberg, M., Berggren, M. & Henningson, D. S. 2002 Linear and nonlinear optimal control in spatial boundary layers. AIAA Paper 2012-2755.Google Scholar
Choi, H., Jeon, W. P. & Kim, J. 2008 Control of flow over a bluff body. J. Fluid Mech. 40, 113139.Google Scholar
Duguet, Y., Schlatter, P., Henningson, D. S. & Eckhardt, B. 2012 Self-sustained localized structures in a boundary-layer flow. Phys. Rev. Lett. 108, 044501.CrossRefGoogle Scholar
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition of pipe flow. Annu. Rev. Fluid Mech. 39, 447468.CrossRefGoogle Scholar
Fransson, J. H. M., Talamelli, A., Brandt, L. & Cossu, C. 2006 Delaying transition to turbulence by a passive mechanism. Phys. Rev. Lett. 96 (6), 064501.Google Scholar
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167197.CrossRefGoogle Scholar
Hammond, E. P., Bewley, T. R. & Moin, P. 1998 Observed mechanisms for turbulence attenuation and enhancement in opposition-controlled wall-bounded flows. Phys. Fluids 10, 24212423.Google Scholar
Herve, A., Sipp, D., Schmid, P. J. & Samuelides, M. 2012 A physics-based approach to flow control using system identification. J. Fluid Mech. 702, 2658.Google Scholar
Hof, B., van Doorne, C. W. H., Westerweel, J., Nieuwstadt, F. T. M., Faisst, H., Eckhardt, B., Wedin, H., Kerswell, R. R. & Waleffe, F. 2004 Experimental observation of nonlinear traveling waves in turbulent pipe flow. Science 305, 15941598.Google Scholar
Huerre, P. & Rossi, M. 1998 Hydrodynamic instabilities in open flows. In Hydrodynamics and Nonlinear Instabilities.Google Scholar
Kim, J. & Bewley, T. R. 2007 A linear systems approach to flow control. Annu. Rev. Fluid Mech. 39, 383417.Google Scholar
Lewis, F. L. & Syrmos, L. V. 1995 Optimal control. Wiley.Google Scholar
Marquet, O., Sipp, D., Chomaz, J.-M. & Jacquin, L. 2008a Amplifier and resonator dynamics of a low-Reynolds-number recirculation bubble in a global framework. J. Fluid Mech. 605, 429443.CrossRefGoogle Scholar
Marquet, O., Sipp, D. & Jacquin, L. 2008b Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech. 615, 221252.CrossRefGoogle Scholar
Monokrousos, A., Bottaro, A., Brandt, L., Di Vita, A. & Henningson, D. S. 2011 Non-equilibrium thermodynamics and the optimal path to turbulence in shear flows. Phys. Rev. Lett. 106, 134502.CrossRefGoogle Scholar
Nagata, M. 1990 Three-dimensional finite amplitude solutions in plane Couette flow. J. Fluid Mech. 217, 519527.Google Scholar
Passaggia, P. Y. & Ehrenstein, U. 2013 Adjoint based optimization and control of a separated boundary-layer flow. Eur. J. Mech. (B/Fluids) 41, 169177.Google Scholar
Pringle, C. C. T. & Kerswell, R. R. 2010 Using nonlinear transient growth to construct the minimal seed for shear flow turbulence. Phys. Rev. Lett. 105, 154502.Google Scholar
Pringle, C. C. T., Willis, A. P. & Kerswell, R. R. 2012 Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos. J. Fluid Mech. 702, 415443.CrossRefGoogle Scholar
Rabin, S. M. E., Caulfield, C. P. & Kerswell, R. R. 2012 Variational identification of minimal seeds to trigger transition in plane Couette flow. J. Fluid Mech. 712, 244272.CrossRefGoogle Scholar
Schmid, P. & Henningson, D. 2001 Stability and Transition in Shear Fows. Springer.CrossRefGoogle Scholar
Schneider, T. M., Eckhardt, B. & Yorke, J. A. 2007 Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett. 99, 034502.Google Scholar
Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57108.Google Scholar
Semeraro, O., Bagheri, S., Brandt, L. & Henningson, D. S. 2011 Feedback control of three-dimensional optimal disturbances using reduced-order models. J. Fluid Mech. 677, 63102.Google Scholar
Shahinfar, S., Sattarzadeh, S. S., Fransson, J. H. M. & Talamelli, A. 2012 Revival of classical vortex generators now for transition delay. Phys. Rev. Lett. 109, 074501.Google Scholar
Skufca, J. D., Yorke, J. A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96, 174101.Google Scholar
Strykowski, P. J. & Sreenivasan, K. R. 1990 On the formation and suppression of vortex ‘shedding’ at low Reynolds numbers. J. Fluid Mech. 218, 71107.Google Scholar
Verzicco, R. & Orlandi, P. 1996 A finite-difference scheme for the three-dimensional incompressible flows in cylindrical coordinates. J. Comput. Phys. 123 (2), 402414.Google Scholar
Viswanath, D. & Cvitanovic, P. 2009 Stable manifolds and the transition to turbulence in pipe flow. J. Fluid Mech. 627, 215233.CrossRefGoogle Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883900.Google Scholar
Waleffe, F. 1998 Three-dimensional states in plane shear flow. Phys. Rev. Lett. 81, 41404143.Google Scholar
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333371.Google Scholar
Zuccher, S., Luchini, P. & Bottaro, A. 2004 Algebraic growth in a blasius boundary layer: optimal and robust control by mean suction in the nonlinear regime. Eur. J. Mech. (B/Fluids) 513, 135160.Google Scholar